The mKdV equation with the
initial value problem is studied numerically by means of the homotopy
perturbation method. The analytical approximate solutions of the mKdV equation
are obtained. Choosing the form of the initial value, the single solitary wave,
two solitary waves and rational solutions are presented, some of which are
shown by the plots.
Cite this paper
Dong, Z. and Wang, F. (2018). Construction of Solitary Wave Solutions and Rational Solutions for mKdV Equation with Initial Value Problem by Homotopy Perturbation Method. Open Access Library Journal, 5, e4383. doi: http://dx.doi.org/10.4236/oalib.1104383.
Chow, S.N., Mallet-Paret, J. and Yorke, J.A. (1978) Finding Zeros of Maps: Homotopy Methods That Are Constructive with Probability One. Mathematics of Computation, 32, 887-899. https://doi.org/10.1090/S0025-5718-1978-0492046-9
Li, T.Y. and Sauser, T. (1987) Homotopy Methods for Eigenvalue Problems. Linear Algebra and Its Applications, 91, 65-74. https://doi.org/10.1016/0024-3795(87)90060-7
Liao, S.J. (1995) An Approximate Solution Technique Not Depending on Small Parameters: A Special Example. International Journal of Non-Linear Mechanics, 30, 371-380. https://doi.org/10.1016/0020-7462(94)00054-E
Ganji, D.D. and Sadighi, A. (2006) Application of He’s Homotopy-Perturbation Method to Nonlinear Coupled Systems of Reaction-Diffusion Equations. International Journal of Nonlinear Sciences and Numerical Simulation, 7, 411-418. https://doi.org/10.1515/IJNSNS.2006.7.4.411
Javidi, M. and Golbabai, A. (2007) A Numerical Solution for Solving System of Fredholm Integral Equations by Using Homotopy Perturbation Method. Applied Mathematics and Computation, 189, 1921-1928. https://doi.org/10.1016/j.amc.2006.12.070
Odibat, Z. (2007) A New Modification of the Homotopy Perturbation Method for Linear and Nonlinear Operators. Applied Mathematics and Computation, 189, 746-753. https://doi.org/10.1016/j.amc.2006.11.188
Abbasbandy, S. (2007) Application of He’s Homotopy Perturbation Method to Functional Integral Equations. Chaos Solitons & Fractals, 31, 1243-1247. https://doi.org/10.1016/j.chaos.2005.10.069
Chen, Z.Y., Huang, N.N., Liu, Z.Z. and Xiao, Y. (1993) An Explicit Expression of the Dark N-Soliton Solution of the MKdV Equation by Means of the Darboux Transformation. Journal of Physics A: Mathematical and General, 26, 1365-1374. https://doi.org/10.1088/0305-4470/26/6/018
Zeng, Y.B. and Dai, H.H. (2001) Constructing the N-Soliton Solution for the mKdV Equation through Constrained Flows. Journal of Physics A: Mathematical and General, 34, L657-L663. https://doi.org/10.1088/0305-4470/34/46/103
Inc, M. (2007) An Approximate Solitary Wave Solution with Compact Support for the Modified KdV Equation. Journal of Applied Mathematics and Computing, 184, 631-637. https://doi.org/10.1016/j.amc.2006.06.062
Wazwaz, A.M. (2007) The Extended Tanh Method for Abundant Solitary Wave Solutions of Nonlinear Wave Equations. Journal of Applied Mathematics and Computing, 187, 1131-1142. https://doi.org/10.1016/j.amc.2006.09.013
Wazwaz, A.M. (2008) New Sets of Solitary Wave Solutions to the KdV, mKdV, and the Generalized KdV Equations. Communications in Nonlinear Science and Numerical Simulation, 13, 331-339. https://doi.org/10.1016/j.cnsns.2006.03.013
Gao, Y.F., Dai, Z.D. and Li, D. (2009) New Exact Periodic Solitary-Wave Solution of mKdV Equation. Communications in Nonlinear Science and Numerical Simulation, 14, 3821-3824. https://doi.org/10.1016/j.cnsns.2008.09.011
Yasar, E. (2010) On the Conservation Laws and Invariant Solutions of the mKdV Equation. Journal of Mathematical Analysis and Applications, 363, 174-181. https://doi.org/10.1016/j.jmaa.2009.08.030
Parkes, E.J. (2010) Observations on the Tanh-Coth Expansion Method for Finding Solutions to Nonlinear Evolution Equations. Journal of Applied Mathematics and Computing, 217, 1749-1754. https://doi.org/10.1016/j.amc.2009.11.037
Zhang, J.B., Zhang, D.J. and Shen, Q. (2011) Bilinear Approach for a Symmetry Constraint of the Modified KdV Equation. Journal of Applied Mathematics and Computing, 218, 4494-4500. https://doi.org/10.1016/j.amc.2011.10.030
Trogdon, T., Olver, S. and Deconinck, B. (2012) Numerical Inverse Scattering for the Kortewegde Vries and Modified Kortewegde Vries Equations. Physica D, 241, 1003-1025. https://doi.org/10.1016/j.physd.2012.02.016
Xin, X.P., Miao, Q. and Chen, Y. (2014) Nonlocal Symmetry, Optimal Systems, and Explicit Solutions of the mKdV Equation. Chinese Physics B, 23, 010203. https://doi.org/10.1088/1674-1056/23/1/010203